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Abstract. We develop various proof techniques for the synthesis of sorting algorithms on binary trees, by extending our previous work on the synthesis of algorithms on lists. Appropriate induction principles are designed and various specific prove-solve methods are experimented, mixing rewriting with assumption-based forward reasoning and goal-based backward reasoning à la Prolog. The proof techniques are implemented in the Theorema system and are used for the automatic synthesis of several algorithms for sorting and for the auxiliary functions, from which we present few here. Moreover we formalize and check some of the algorithms and some of the properties in the Coq system.

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The construction of a theory for binary trees is presented, based on the systematic exploration of the properties necessary for the proof-based synthesis and certification of sorting algorithms for binary trees. The process is computer supported, being realised in the frame of the Theorema system, with some additional proofs in Coq required for algorithm certification. The result of the exploration consists in 11 definitions, 3 axioms, and more than 200 properties. Also, more than 5 algorithms for sorting binary trees are generated.

We develop logic and combinatorial methods for automating the generation of sorting algorithms for binary trees, starting from input-output specifications and producing conditional rewrite rules. The main approach consists in proving (constructively) the existence of an appropriate output from every input. The proof may fail if some necessary sub-algorithms are lacking. Then, their specifications are suggested and their synthesis is performed by the same principles.Our main goal is to avoid the possibly prohibitive cost of pure resolution proofs by using a natural-style proving in which domain-specific strategies and inference steps lead to a significant increase of efficiency. In addition to classical techniques for natural-style proving, we introduce novel ones (priority of certain types of assumptions, transformation of elementary goals into conditions, special criteria for decomposition of the goal and of the assumptions), as well as methods based on the properties of domain-specific relations and functions. In particular, we use combinatorial techniques in order to generate possible witnesses, which in certain cases lead to the discovery of new induction principles. From the proof, the algorithm is extracted by transforming inductive proof steps into recursions, and case-based proof steps into conditionals.The approach is demonstrated in parallel using the Theorema system, by developing the theory, implementing the prover, and performing the proofs of the necessary properties and synthesis conjectures. It is also validated in the Coq system, which allows to compare the facilities of the two systems from the point of view of our application.

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